Some spectral relations

We consider the zeta function of a second-order differential operator which has a secon-dorder turning point: $$Lu = \frac{{d^2 u}}{{dx^2 }} + [\lambda ^2 q(x) + R(x)]u,$$ where q(x)=x²q₁(x), q₁(x)≠0 and u (0)=u (1)=0. We construct an asymptotic series and calculate regularized traces for the eigenvalues of this operator.     

Filtering in Legendre spectral methods

We discuss the impact of modal filtering in Legendre spectral methods, both on accuracy and stability. For the former, we derive sufficient conditions on the filter to recover high order accuracy away from points of discontinuity. Computational results confirm that less strict necessary conditions appear to be adequate. We proceed to discuss a instability mechanism in polynomial spectral methods and prove that filtering suffices to ensure stability. The results are illustrated by computational experiments.

     

谱逼近的某些新进展

本文综述谱逼近的某些新进展,非线怀计算不稳定性,数据的不连续性和解的奇异性会破坏谱方法的高精度,各种滤波方法,本质不振荡多项式插值,正交逼近的重构造方法和某些Ｈｉｌｂｅｒｔ空间中的Ｊａｃｏｂｉ逼近被应用,它们使得谱方法更有效。     

Some spectral mapping theorems through local spectral theory

The spectral mapping theorems for Browder spectrum and for semi-Browder spectra have been proved by several authors [14], [29] and [33], by using different methods. We shall employ a local spectral argument to establish these spectral mapping theorems, as well as, the spectral mapping theorem relative to some other classical spectra. We also prove that ifT orT* has the single-valued extension property some of the more important spectra originating from Fredholm theory coincide. This result is extended, always in the caseT orT* has the single valued extension property, tof(T), wheref is an analytic function defined on an open disc containing the spectrum ofT. In the last part we improve a recent result of Curto and Han [10] by proving that for every transaloid operatorT a-Weyl’s theorem holds forf(T) andf(T)*. The research was supported by the International Cooperation Project between the University of Palermo (Italy) and Conicit-Venezuela.     

Minimal perturbation approach implementation in spectral methods

Pseudospectra and minimal perturbation approach (MPA) principles are used and implemented to provide more stability and more rigorous mathematical foundations for the functional expansion multi-domain spectral method (MDSM). The implemented formulation is used to analyze optical waveguides and semiconductor quantum wells. The results show that MDSM formulation based on the MPA is very stable and slightly more accurate as compared to the Tau MDSM.     

Recent advances in Krylov subspace spectral methods

This paper reviews the main properties, and most recent developments, of Krylov subspace spectral (KSS) methods for time-dependent variable-coefficient PDE. These methods use techniques developed by Golub and Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral domain in order to achieve high-order accuracy in time and stability characteristic of implicit time-stepping schemes, even though KSS methods themselves are explicit. In fact, for certain problems, 1-node KSS methods are unconditionally stable. Furthermore, these methods are equivalent to high-order operator splittings, thus offering another perspective for further analysis and enhancement. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some spectral properties of differential operators

Mathematical Notes -     

Some examples inL p spectral geometry

We construct surfaces \(T_{x_0 }^p \) with volume, first eigenvalue, andL ^p norm of the curvature bounded independent ofx ₀, but whose Cheeger constant tends to 0 asx ₀ tends to infinity.     

Some remarks on spectral states ofLMC-algebras

We generalize the notion of a spectral state (as introduced for Banach algebras by Moore, Bonsall and Duncan) to the context of locally multiplicatively-convex (LMC) algebras by proceeding in a way analogous to the generalization of numerical range theory from Banach algebras toLMC-algebras carried out by Giles and Koehler. Among the results obtained in this note are integral representations of spectral states by probability measures on the structure space ofA and the determination of the extreme points of the convex set\(\Omega _A \) of all spectral states on a commutativeLMC-algebraA (which is related to different Choquet boundaries) as well as a characterization of symmetric involutions by the coincidence of the notions of positive state and spectral state and a characterization of theQ-property by the weak-^*-boundedness of\(\Omega _A \). The paper finishes with two elementary commutativity criteria involving spectral states and two Korovkin-type theorems for the approximation of unital algebra homomorphisms by σ-equicontractive nets of linear operators mapping anLMC-algebraA into theLMC-algebra of all continuous complex-valued functions on a completely regular spaceX.     

Some spectral properties ofm-accretive differential operators

Form-accretive operatorT=T ₀+q compactness of its resolventR(z, T), zP(T) and ofR(z, T)–R(z, T ₀),zP(T)P(T ₀) under suitable assumptions onT ₀ and their perturbationq is established. This result is used in the study of spectral properties ofT andT ₀.     

Relaxation schemes for spectral multigrid methods

The effectiveness of relaxation schemes for solving the systems of algebraic equations which arise from spectral discretizations of elliptic equations is examined. Iterative methods are an attractive alternative to direct methods because Fourier transform techniques enable the discrete matrix-vector products to be computed almost as efficiently as for corresponding but sparse finite difference discretizations. Preconditioning is found to be essential for acceptable rates of convergence. Preconditioners based on second-order finite difference methods are used. A comparison is made of the performance of different relaxation methods on model problems with a variety of conditions specified around the boundary. The investigations show that iterations based on incomplete LU decompositions provide the most efficient methods for solving these algebraic systems.     

Filter-based stabilization of spectral element methods

We present a simple filtering procedure for stabilizing the spectral element method (SEM) for the unsteady advection–diffusion and Navier–Stokes equations. A number of example applications are presented, along with basic analysis for the advection–diffusion case.     

Some results on the spectral reconstruction problem

This paper is concerned with the spectral version of the reconstruction conjecture: Whether a graph with n>2 vertices is determined (up to isomorphism) by the collection of its spectrum and the spectrum of its vertex-deleted graphs? Some positive results as well as a method for constructing counterexamples to the problem are provided.     

Some notes on the adams spectral sequence

In the beginning of 1980's Cohen, R. proved thath0b(1k)=B(1,k)=B(1,k)⊗ζ₁ survives toE _∞ in the Adams spectral sequence. Later, Cohen, R. and Goerss, P. proved that is a permanent cycle. And they are represented by ζ _k ,ηj respectively. Here the author proved: Theorem 2: Let 2≤s≤p−1,j≥3, then β _s η _j ≠0. Theorem 3: For 2≤s≤p−1,k≥2, β _s ζ _k ≠0 in the stable homotopy groups of spheres. As a remark, we get in . Supported by Doctoral Program Fundation.     

Some asymptotic spectral properties of singular operators

A class of uniformly elliptic, positive operators in Rⁿ with discrete spectrum is considered for which the coefficients of the derivatives of even order and the free term increase at the same rate, while the other coefficients play a subordinate role. The first term of the asymptotic expansion of the spectral function and N () is found for such operators; here N () , where the _n are the eigenvalues of the operator.Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 169–172, August, 1968.